**Memoirs of the American Mathematical Society**

2017;
101 pp;
Softcover

MSC: Primary 03;

**Print ISBN: 978-1-4704-2657-6
Product Code: MEMO/249/1187**

List Price: $75.00

AMS Member Price: $45.00

MAA Member Price: $67.50

**Electronic ISBN: 978-1-4704-4141-8
Product Code: MEMO/249/1187.E**

List Price: $75.00

AMS Member Price: $45.00

MAA Member Price: $67.50

# Induction, Bounding, Weak Combinatorial Principles, and the Homogeneous Model Theorem

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*Denis R. Hirschfeldt; Karen Lange; Richard A. Shore*

Goncharov and Peretyat'kin independently gave necessary and sufficient conditions for when a set of types of a complete theory \(T\) is the type spectrum of some homogeneous model of \(T\). Their result can be stated as a principle of second order arithmetic, which is called the Homogeneous Model Theorem (HMT), and analyzed from the points of view of computability theory and reverse mathematics. Previous computability theoretic results by Lange suggested a close connection between HMT and the Atomic Model Theorem (AMT), which states that every complete atomic theory has an atomic model. The authors show that HMT and AMT are indeed equivalent in the sense of reverse mathematics, as well as in a strong computability theoretic sense and do the same for an analogous result of Peretyat'kin giving necessary and sufficient conditions for when a set of types is the type spectrum of some model.

#### Table of Contents

# Table of Contents

## Induction, Bounding, Weak Combinatorial Principles, and the Homogeneous Model Theorem

- Cover Cover11
- Title page i2
- Chapter 1. Introduction 18
- Chapter 2. Definitions 1522
- Chapter 3. The Atomic Model Theorem and Related Principles 2128
- Chapter 4. Defining Homogeneity 2734
- Chapter 5. Closure Conditions and Model Existence 4148
- Chapter 6. Extension Functions and Model Existence 5158
- Chapter 7. The Reverse Mathematics of Model Existence Theorems 7582
- Chapter 8. Open Questions 8794
- Appendix A. Approximating Generics 8996
- Appendix B. Atomic Trees 93100
- Appendix C. Saturated Models 97104
- Bibliography 99106
- Back Cover Back Cover1114